ON OSCILLATION AND ASYMPTOTIC STABILITY OF A DELAY DIFFERENTIAL EQUATION WITH RICHARD S NONLINEARITY

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1 ON OSCILLATION AND ASYMPTOTIC STABILITY OF A DELAY DIFFERENTIAL EQUATION WITH RICHARD S NONLINEARITY Leonid Berezansky 1 1 Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel Lev Idels 2 Mathematics Department, Malaspina University-College, 900 Fifth Street Nanaimo, BC V9R 5S5, Canada Abstract For a scalar nonlinear delay differential equation [ dn m dt = rtnt γ ] a b k Ng k t, g k t t, sufficient conditions for oscillation of all solutions and asymptotical stability of the positive equilibrium are obtained. 1. INTRODUCTION Consider the following logistic differential equation which is widely used in Population Dynamics dn 1 dt = rn N. K Here Nt is the size of a population, r 0 is an intrinsic growth rate, K is a carrying capacity or saturation level. A variety of nonlinear differential equations has been developed to construct numerous models of Mathematical Biology [1], [2], [3]. In order to model processes in nature and engineering it is frequently required to know system states in the past. Depending on the phenomena under study the after-effects represent duration of some hidden processes. In general, DDE s exhibit much more complicated dynamics than ODE s since a time lag can change a stable equilibrium into an unstable one and make populations fluctuate, they provide a richer mathematical framework compared with ordinary differential equations for the analysis of biosystems dynamics. Introduction of complex models of Population Dynamics, based on nonlinear DDE s, has received much attention in the literature in recent years. The application of delay equations to biomodelling is in many cases associated with studies of dynamic phenomena like oscillations, bifurcations, and chaotic behavior. Time delays represent an additional level of complexity that can be incorporated in a more detailed analysis of a particular system.

2 Delay logistic equation dn dt = rn 1 N τ. 1 K appeared in 1948 in Hutchinson s paper [4]. Here N τ Nt τ, τ > 0. Autonomous equation 1 has been extensively investigated by numerous authors. The first paper on the oscillation of a nonautonomous logistic delay differential equation was published in [5]. Since this publication, the oscillation of the logistic DDE as well as its generalizations were studied by many mathematicians. Some of these results can be found in the monographs [6], [7], [8]. It is a well-known fact, that the traditional logistic model in some cases produces artificially complex dynamics, therefore it would be reasonable to get away from the specific logistic form in studying population dynamics and use more general classes of growth models. For example, in order to drop an unnatural symmetry of the logistic curve, we consider the modified logistic form of Pella and Tomlinson [9], [10] or Richards growth equation with delay [ dn γ ] dt = rn Nτ 1. 2 K According to [9], 0 < γ < 1 for invertebrate populations examples of invertebrates are insects, worms, starfish, sponges, squid, plankton, crustaceans, and mollusks, and γ 1 for the vertebrate populations these include amphibians, birds, fish, mammals, and reptiles. In [11] the authors considered Eq.2 with several delays. They obtained conditions for existence of positive solutions and studied so-called long time average stability. In this paper we obtain oscillation and local stability results for nonautonomous Eq.2 with several delays. 2. PRELIMINARIES Our object is a scalar nonlinear delay differential equation [ m γ ] Ṅt = rtnt a b k Ng k t, t 0, 3 under the following conditions: a1 rt is a Lebesgue measurable essentially bounded on [0, function, rt 0. a2 g k : [0, R are Lebesgue measurable functions, g k t t, lim g k t =, k = 1,..., m. a3 a > 0, b k > 0, γ > 0. Together with 3 we consider for each t 0 0 an initial value problem [ m γ ] Ṅt = rtnt a b k Ng k t, t t 0, 4 Nt = ϕt, t < t 0, Nt 0 = N 0. 5 We also assume that the following hypothesis holds a4 ϕ :, t 0 R is a Borel measurable bounded function, ϕt 0, N 0 > 0. 2

3 Definition. A locally absolutely continuous function x : R R is called a solution of problem 4, 5, if it satisfies equation 4 for almost all t [t 0, and equalities 5 for t t 0. Lemma 1 [11] Suppose Conds.a1-a4 hold for equation has a unique positive solution Nt, t t 0. Then the problem 3. OSCILLATION CRITERIA Definition. We say that a function yt is nonoscillatory about a number K if yt K is eventually positive or eventually negative. Otherwise yt is oscillatory about K. Eq.3 has a positive equilibrium N = a γ 1 m. In this section we study oscillation b k of solutions of 3 about N. We will present here some lemmas which will be used in this section. Consider the linear delay differential equation and the differential inequalities ẋt + r k txh k t = 0, t 0, 6 ẋt + r k txh k t 0, t 0, 7 ẋt + r k txh k t 0, t 0. 8 Lemma 2 [6] Let a1-a2 hold for the parameters of Eq.6. Then the following statements are equivalent: 1. There exists a non-oscillatory solution of equation There exists an eventually positive solution of the inequality There exists an eventually negative solution of the inequality 8. Lemma 3 [6] Let a1-a2 hold for the parameters of Eq.6. If lim inf t max k h k t then all the solutions of equation 6 are oscillatory. r i sds > 1/e, 9 i=1 Theorem 1 Suppose a1-a4 hold and rsds =. 10 Then for every nonoscillatory solution Nt of 3 we have 0 lim Nt = N. 11 3

4 Proof. After a substitution Eq.3 reduced to the following equation where B k = Nt = N 1 + xt 12 [ m γ ] ẋt = art1 + xt B k 1 + xg k t 1, t 0, 13 b k m i=1 b i. It is evident, that m B k = 1. The zero solution is an equilibrium of Eq.13, which suits to the equilibrium N of Eq.3. By Lemma 1 any solution of 3 is positive. Then for any solution of 13 we have 1 + xt > 0. To prove the theorem we have to show that for every nonoscillatory about zero solution of 13 we have lim xt = Suppose xt is a nonoscillatory solution of 13. Without loss of generality we can assume that xt > 0, t 0. Hence m γ m γ B k 1 + xg k t 1 B k 1 = 0. Then ẋt 0 and hence there exists Suppose l > 0. Equality 13 implies lim xt = l. [ t m γ ] xt = x0 a rs1 + xs B k 1 + xg k s 1 ds If t + then the right hand side of 15 tends to, the left hand side has a finite limit. This contradiction proves the theorem. Theorem 2 Suppose conditions a1-a4 and 10 hold. If γ > 1 and there exists ɛ > 0 such that all solutions of linear differential equation ẏt = aγrt1 ɛ B k yg k t 16 are oscillatory, then all solutions of 3 are oscillatory about N. Proof. It is sufficient to prove, that all solutions of 13 are oscillatory about zero. Suppose the exists a nonoscillatory solution x of 13. Without loss of generality we can assume, that xt > 0, t 0. Theorem 1 implies, that for some t 0 > 0 and for t t 0 we have 0 < xt < ɛ. 4

5 Consider the following function m γ fu 1,..., u m = B k 1 + u k 1 γ B k u k. We have Hence f0,..., 0 = 0, f m γ 1 = γ B k 1 + u k B k γb k, u k m 2 f u i u j = γγ 1 Taylor s Formula implies that where f u k 0,..., 0 = 0, γ 2 B k 1 + u k B i B j. 2 f u i u j 0,..., 0 = γγ 1B i B j. fu 1,..., u m = γγ 1 B i B j u i u j + o u, i=1 j=1 m 2 ot u = u k, lim = 0. t 0 t Then for u k 0, k = 1,..., m and u sufficiently small fu 1,..., u m 0. Hence for ɛ small enough we have ẋt aγrt1 ɛ B k xg k s, t 0. Lemma 2 implies now that Eq.16 has a nonoscillatory solution. We have a contradiction with our assumption. The theorem is proven. Corollary 2.1 Suppose conditions a1-a4and 10 hold, γ > 1, lim inf aγ t max k g k t rsds > 1/e. 17 Then all solutions of 3 are oscillatory about N. Proof Inequality 17 implies, that for some ɛ > 0 lim inf aγ1 ɛ t max k g k t B i rsds > 1/e. i=1 Lemma 3 and Theorem 2 imply this corollary. 4. ASYMPTOTIC STABILITY Consider a general nonlinear delay differential equation ẋt = ft, xt, xg 1 t,..., xg m t, t 0, 18 with the initial function and the initial value xt = ϕt, t < 0, x0 = x 0, 19 5

6 under the following conditions: b1 ft, u 0, u 1,..., u m satisfies Caratheodory conditions: it is Lebesgue measurable in the first argument and continuous in other arguments, ft, 0,..., 0 = K; b2 g k t are Lebesgue measurable functions, g k t t, sup[t g k t] < ; t 0 b3 ϕ :, 0 R is a Borel measurable bounded function. We will assume that the initial value problem has a unique global solution xt, t 0. Definition. We will say that the equilibrium K of Eq.18 is locally stable, if for any ɛ > 0 there exists δ > 0 such that for every initial conditions x0 < δ 0, ϕt < δ 0, δ 0 δ, for the solution xt of we have xt K < ɛ, t 0. If, in addition, lim xt K = 0, then the equilibrium K of Eq.18 is locally asymptotically stable. Suppose there exist M > 0, γ > 0 such that xt K M exp{ γt} x0 + sup ϕt t<0 for all x0 and ϕt such that x0 + sup t<0 ϕt is sufficiently small. Then we will say that the equilibrium K of Eq.18 is exponentially stable. Lemma 4 [12] Suppose a1, b2 hold for linear equation 6 and lim sup Then Eq.6 is exponentially stable. r k tt h k t < 1. Lemma 5 [13], [14] Suppose that for sufficiently small u if u k u, k = 0,..., m then F ft, u 0,..., u m t, K,..., K u k = ou, u k where lim u 0 ou u = 0. If the linear equation ẏt = k=0 k=0 F u k t, 0,..., 0 yg k t is exponentially stable, then the the equilibrium K of Eq.18 is locally asymptotically stable. Theorem 3 Suppose that for equation 3 Conds. a1, b2, a3, a4 hold and lim sup Then equilibrium N of Eq.3 is asymptotically stable. aγrt B k t g k t <

7 Proof. A substitution Nt = N 1 + xt implies that equilibrium N of Eq.3 is asymptotically stable if and only if the zero solution of 13 is asymptotically stable. Lemma 4 and inequality 20 imply that linear equation ẋt = aγrt B k xg k t is exponentially stable. Lemma 5 implies now that the zero solution of 13 is asymptotically stable. References [1] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, [2] M. Kot, Elements of Mathematical Ecology, Cambridge Univ. Press, [3] C. T. H. Baker, Retarded Differential Equations, J. Comp. Appl. Math., , [4] G.E. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci. 50, pp [5] B.G. Zhang and K. Gopalsamy, Oscillation and nonoscillation in a nonautonomous delay-logistic equation, Quart. Appl. Math. XLVI 1988, [6] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations, 1991, Clarendon Press, Oxford. [7] K. Gopalsamy, Stability and Oscillation in Delay Differential Equations of Population Dynamics, 1992, Kluwer Academic Publishers, Dordrecht, Boston, London. [8] L.N. Erbe, Q. Kong and B.G. Zhang, Oscillation Theory for Functional Differential Equations, 1995, Marcel Dekker, New York, Basel. [9] A. Tsoularis and J. Wallace, Analysis of logistic growth models, Mathematical Biosciences, , [10] J. Pella and P. Tomlinson, A Generalized Stock-Production Model, Inter.-Am. Trop. Tuna Comm. Bull., 13,1969, [11] Jose J. Miguel, A. Ponosov, A. Shindiapin, On a delay equation with Richards nonlinearity. Proceedings of the Third World Congress of Nonlinear Analysts, Part 6 Catania, Nonlinear Anal , no. 6, [12] T. Krisztin, On stability properties for one-dimensional functional-differential equations. Funkcial. Ekvac , no. 2, [13] R. Bellman; K. Cooke, Differential-difference equations. Academic Press, New York-London 1963, 462 pp. 7

8 [14] V. Kolmanovskii, A. Myshkis, Introduction to the theory and applications of functional-differential equations. Mathematics and its Applications, 463. Kluwer Academic Publishers, Dordrecht, pp. 8